Spectral properties of the discrete random displacement model

Abstract

We investigate spectral properties of a discrete random displacement model, a Schrödinger operator on ${\ell }^2({\mathbb{Z}}^d)$ with potential generated by randomly displacing finitely supported single-site terms from the points of a sublattice of ${\mathbb{Z}}^d$. In particular, we characterize the upper and lower edges of the almost sure spectrum. For a one-dimensional model with Bernoulli distributed displacements, we can show that the integrated density of states has a $1/{\log }^2$-singularity at external as well as internal band edges.